Question

Question #4697e

Answer 1

`4=E`
(Note that my pictures are hugely off scale)

Explanation:

This may be one of the problems that requires some thinking before trying.

Let's label what we know. Here, I have created a picture in Paint.

We are trying to solve for `E`.

Hmm. Let's try to label the angles.
We know that the angles `d` and `b` are right angles.
Let's name angle `c` as `x`.
We now have:

Using the fact that the angles of a triangle always add up to 180 degrees, we can figure out angle `a` in terms of `x`.
`90+x+m/_c=180`
=>`x+m/_c=90`
=>`m/_c=90-x`
We now have:

Similarly, we can figure out the missing angles for triangles `adb` and `bdc`

The missing angle for triangle `adb`, `m`, is equal to `180-90-(90-x)`
=>`180-90-(90-x)=m`
=>`90-90+x=m`
=>`x=m`

Similarly, the missing angle for triangle `bdc`, `M` is equal to `180-90-x`
=>`180-90-x=M`
=>`90-x=M`

We label these angles to get the following:

We see that the triangles `adb` and `bdc` are similar using the AAA similarity theorem.(The corresponding angles are congruent)

Basically, we have separated our original triangle to two pieces:

We can now write a proportion. (The ratio of a pair of corresponding sides are equal to other pairs)

We have:
`8/E=E/2`

We can now solve for `E`.
=>`16=E^2`
=>`sqrt16=E`
=>`4=E`